Finite Group Databasebeta

← #20122

Subgroups — #20122

Order 256.

304 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3366C211yesmaximal, characteristic, complemented
2#3357C211yesmaximal, characteristic, complemented
3#3341C211yesmaximal, characteristic, complemented
4#3305C211yesmaximal, characteristic, complemented
5#1722C211yesmaximal, characteristic, complemented
6#1697C211yesmaximal, characteristic, complemented
7#1696C211yesmaximal, characteristic, complemented
8#1677C211yesmaximal, characteristic, complemented
9#1675C211yesmaximal, characteristic, complemented
10#1634C211yesmaximal, characteristic, complemented
11#1632C211yesmaximal, characteristic, complemented
12#1631C211yesmaximal, characteristic, complemented
13#1624C211yesmaximal, characteristic, complemented
14#1586C211yesmaximal, characteristic, complemented
15#1585C211yesmaximal, characteristic, complemented
16#555C2211yescharacteristic
17#555C2211yescharacteristic
18#554C2211yescharacteristic
19#550C2211yescharacteristic
20#500C2211yescharacteristic
21#500C2211yescharacteristic
22#494C2211yescharacteristic
23#492C2211yescharacteristic
24#490—21no—
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#485—21no—
28#482—21no—
29#482—21no—
30#482—21no—
31#480—21no—
32#478—21no—
33#477C2211yescharacteristic
34#477C2211yescharacteristic
35#476C2211yescharacteristic
36#476C2211yescharacteristic
37#476C2211yescharacteristic
38#476C2211yescharacteristic
39#468—21no—
40#466—21no—
41#466—21no—
42#459—21no—
43#459C2211yescharacteristic
44#459C2211yescharacteristic
45#449—21no—
46#446C2211yescharacteristic
47#442—21no—
48C4×SD16C2211yescharacteristic
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic
51C8×D4C2211yescharacteristic
52C8×D4C2211yescharacteristic
53#428—21no—
54#427—21no—
55#416—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#333C2211yescharacteristic
60#333C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×Q16—41no—
69C2×SD16—41no—
70C2×SD16D411yescharacteristic
71C2×D8—21no—
72C2×D8—21no—
73C2×D8D411yescharacteristic
74C2×M16—21no—
75C22×C8—21no—
76C22×C8—21no—
77C4⋊Q8C2311yescharacteristic
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80#127—21no—
81#127C2311yescharacteristic
82#125—21no—
83#124—41no—
84#124—41no—
85C22⋊Q8—21no—
86C22⋊Q8—41no—
87#121—21no—
88#121—21no—
89#121—21no—
90#121—21no—
91#121—21no—
92#121—41no—
93C4×Q8C2311yescharacteristic
94C4×D4—21no—
95C4×D4—21no—
96C4×D4C2311yescharacteristic
97C4×D4C2311yescharacteristic
98C4×D4C2311yescharacteristic
99#117—21no—
100#117—21no—
101#117—21no—
102#117—21no—
103C2×C4⋊C4—21no—
104#107—21no—
105#107—21no—
106#107—21no—
107#107—21no—
108#107—21no—
109#107—21no—
110#107D411yescharacteristic
111#106—21no—
112#106—21no—
113#106—21no—
114#106—21no—
115#106D411yescharacteristic
116C4⋊C8—21no—
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121#103—21no—
122#103—21no—
123#103D411yescharacteristic
124#102—21no—
125#102—21no—
126#102—21no—
127#102—21no—
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102D411yescharacteristic
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139C22⋊C8—21no—
140#97—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—81no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8C2×D411yescharacteristic
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4C2×D411yescharacteristic
158C2×D4C2×D411yescharacteristic
159C2×D4C2×D411yescharacteristic
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—41no—
164C22×C4—41no—
165Q16—82no—
166SD16—41no—
167SD16—82no—
168D8—41no—
169D8—42no—
170D8—81no—
171M16—42no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—42no—
178C2×C8—81no—
179C2×C8C2×D411yescharacteristic
180C2×C8C2×D411yescharacteristic
181C2×C8C2×D411yescharacteristic
182C2×C8C2×D411yescharacteristic
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4C2×D411yescharacteristic
193C4⋊C4C2×D411yescharacteristic
194C4⋊C4C2×D411yescharacteristic
195C4⋊C4C2×D411yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—21no—
198C22⋊C4—21no—
199C22⋊C4—41no—
200C22⋊C4—41no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C42—21no—
206C42—41no—
207C42—41no—
208C42C2411yescharacteristic
209C23—21no—
210C23—21no—
211C23—21no—
212C23—41no—
213Q8—21no—
214Q8—21no—
215Q8—81no—
216Q8—81no—
217Q8—81no—
218D4—21no—
219D4—21no—
220D4—21no—
221D4—21no—
222D4—21no—
223D4—21no—
224D4—41no—
225D4—41no—
226D4—41no—
227D4—41no—
228D4—81no—
229D4—82no—
230D4—161no—
231D4—161no—
232D4—161no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—82no—
248C2×C4—161no—
249C2×C4C22×D411yescharacteristic
250C2×C4C22×D411yescharacteristic
251C2×C4H(5,𝔽2)11yescharacteristic
252C8—22no—
253C8—21no—
254C8—21no—
255C8—21no—
256C8—21no—
257C8—41no—
258C8—41no—
259C8—41no—
260C8—41no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—81no—
271C22—81no—
272C22—81no—
273C22D4211yescharacteristic
274C4—21no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—81no—
281C4—81no—
282C4—81no—
283C4#57511yescharacteristic
284C4#57511yescharacteristic
285C4#57711yescharacteristic
286C4#57811yescharacteristic
287C2—81no—
288C2—81no—
289C2—81no—
290C2—161no—
291C2#326711yescharacteristic
292C2#326911yescharacteristic
293C2#329211yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3366C2semidirect
2. #3357C2semidirect
3. #3341C2semidirect
4. #3305C2semidirect
5. #1722C2semidirect
6. #1697C2semidirect
7. #1696C2semidirect
8. #1677C2semidirect
9. #1675C2semidirect
10. #1634C2semidirect
11. #1632C2semidirect
12. #1631C2semidirect
13. #1624C2semidirect
14. #1586C2semidirect
15. #1585C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#33662
#33572
#33412
#33052
#17222
#16972
#16962
#16772
#16752
#16342
#16322
#16312
#16242
#15862
#15852